The possibility to deposit purely organic and hybrid inorganic–organic materials in a way parallel to the state-of-the-art gas-phase deposition method of inorganic thin films, i.e., atomic layer deposition (ALD), is currently experiencing a strongly growing interest. Like ALD in case of the inorganics, the emerging molecular layer deposition (MLD) technique for organic constituents can be employed to fabricate high-quality thin films and coatings with thickness and composition control on the molecular scale, even on complex three-dimensional structures. Moreover, by combining the two techniques, ALD and MLD, fundamentally new types of inorganic–organic hybrid materials can be produced. In this review article, we first describe the basic concepts regarding the MLD and ALD/MLD processes, followed by a comprehensive review of the various precursors and precursor pairs so far employed in these processes. Finally, we discuss the first proof-of-concept experiments in which the newly developed MLD and ALD/MLD processes are exploited to fabricate novel multilayer and nanostructure architectures by combining different inorganic, organic and hybrid material layers into on-demand designed mixtures, superlattices and nanolaminates, and employing new innovative nanotemplates or post-deposition treatments to, e.g., selectively decompose parts of the structure. Such layer-engineered and/or nanostructured hybrid materials with exciting combinations of functional properties hold great promise for high-end technological applications.